Suppose there are two potential pools of customers for the gym. One group gets a high benefit from joining, but a smaller benefit from actually working out. The other group gets a smaller benefit from joining, but a larger benefit from working out. Specifically, suppose Group i has preferences given by: T(q) = θi + θiVi(q) − T if the consumer buys q > 0, 0 if he or she does not buy, where Vi(q) = (1− (1− q θi)^2)/2 . T is the tariff the firm charges. Assume it could have a fixed and variable component (two-part tariff). Assume θ1 < θ2 (here θ2 likes to join but doesn’t like working out so much, while θ1 is the reverse. Assume λ is the proportion of the population in Group 1.
A. Derive the demand for each group assuming each decides to consume a positive amount. Derive each consumer’s surplus before the fixed fee for a given variable price p.
B. How will the monopolist price under third-degree price discrimination? What will monopolist profit be?
C. Suppose the monopolist is constrained to offering no fixed fee and one variable price to both firms. What will the price be? What will the monopolist’s profit be? How about the consumer surplus?
D. Suppose the monopolist cannot observe who is in which group. What is the optimal two-part tariff the monopolist can offer? (This may depend on the relationship between θ1 and θ2; at a given variable price p it is not always a Group 1 or Group 2 consumer who has a higher surplus).